New Perspectives on the Relativistically Rotating Disk

and Non-time-orthogonal Reference Frames

Robert D. Klauber

1100 University Manor Dr, 38B

Fairfield, Iowa 52556

rklauber@netscape.net

October 1998

Abstract

The rotating disk problem is analyzed on the premise that proper interpretation of ex-perimental evidence leads to the conclusion that the postulates upon which relativity theoryis based, particularly the invariance of the speed of light, are not applicable to rotatingframes. Different postulates based on the Sagnac experiment are proposed, and from thesepostulates a new relativistic theory of rotating frames is developed following steps similarto those initially followed by Einstein for rectilinear motion. The resulting theory agreeswith all experiments, resolves problems with the traditional approach to the rotating disk,and exhibits both traditionally relativistic and non-relativistic characteristics. Of particularnote, no Lorentz contraction exists on the rotating disk circumference, and the disk surface,contrary to the assertions of Einstein and others, is found to be Riemann flat. The variablespeed of light found in the Sagnac experiment is then shown to be characteristic of non-time-orthogonal reference frames, of which the rotating frame is one. In addition, the widelyaccepted postulate for the equivalence of inertial and non-inertial standard rods with zerorelative velocity, used liberally in prior rotating disk analyses, is shown to be invalid for suchframes. Further, the new theory stands alone in correctly predicting what was heretoforeconsidered a ”spurious” non-null effect on the order of 10−13 found by Brillet and Hall in themost accurate Michelson-Morley type test to date. The presentation is simple and pedagogicin order to make it accessible to the non-specialist.

Key words: relativistic, rotating disk, Sagnac, rotating frame, non-time-orthogonal frame.

1 BACKGROUND

1.1 Perspectives of Einstein and Others

Albert Einstein never published a technical paper directly addressing the problem of the rel-ativistically rotating rigid disk, although in private writings[1], in three books for the generalaudience[2],[3],[4], and as support for the use of generalized coordinates in his landmark 1916paper[5], he purported that the space of such a rotating disk is curved, not flat. He further

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attributed his early insights into general relativity theory to be a direct result of contemplatingthe curvature of such a rotating system. His perspective on the problem is revealed in a private1919 letter found by Stachel[6] in the Einstein Archives at the Institute for Advanced Study inPrinceton, New Jersey.

In that letter Einstein considers measuring rods laid out along the disk’s radii and circum-ference and assumes Lorentz contraction exists along the circumference due to the tangentialvelocity v= ωr of the disk at a given radius r. In Einstein’s words:

....imagine a ”snapshot” taken from [the non-rotating frame] ... On this snapshot the radialmeasuring rods have the length l, the tangential ones, however, the length l(1−v2/c2)1/2. The”circumference” [therefore is] U = [2πr ]/(1 - v2/c2)1/2.

He repeated this type of reasoning elsewhere [2, 3, 4], and used it to conclude that therotating disk surface is not Euclidean since U > 2πr.

But not everyone agrees. In a 1951 letter[7] Einstein noted that Eddington and Lorentzconsidered the geometry on the disk to be flat, and he stated that he did not know what theymeant. No references recounting Eddington’s and Lorentz’s thinking on the subject seem to beavailable, but others, such as Levy[8], appear to agree with them. Strauss[9] concludes that thespace is curved, but argues that Einstein’s logic was flawed, noting that

If the measuring rods laid along the circumference of the rotating disk are Lorentz contractedwith respect to the inertial frame, so are the distances on the circumference they are supposedto measure; hence the two effects would cancel each other, and the ratio U/D would turn out toequal π as in the Euclidean plane.

Grøn [10], however, citing Møller [11] and Landau and Lifshitz [12], contends that thisargument is wrong, Weber [13] supports Grøn’s view, and Stachel [1] effectively concedes thepoint to Einstein.

1.2 Relevant Relativity Principles

Special relativity is restricted to inertial systems and is derived from two symmetry postulates:

1. The speed of light is the same for all inertial observers (it is invariant) and equals c.2. There is no preferred inertial reference frame. (Velocity is relative, and the laws of nature

are covariant, i.e., the same for all inertial observers.)

General relativity is applicable to non-inertial systems and is based on generalizations of theabove two postulates embellished with other certain principles/assumptions, including:

1. The speed of light is invariant and equals c for non-inertial observers provided that it ismeasured locally by local standard clocks and measuring rods [14].

2. There is no preferred non-inertial frame. (The laws of nature are also covariant fornon-inertial observers, although coordinate metrics different from those of special relativity areneeded to represent those laws.)

3. Gravity and acceleration are locally indistinguishable, i.e., the equivalence principle. (Overfinite distances, gravity can, however, be distinguished from acceleration due to the presence ofgravitational tidal forces or geodesic deviation.)

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4. Neither gravity nor acceleration changes the length of a standard rod or the rate of astandard clock relative to a nearby freely falling (inertial) standard rod or standard clock havingthe same velocity. (This is an assumption rarely emphasized in most texts, though Møller[11]makes the point clearly, and Einstein [15] emphasized it a number of times.) We will call thisthe ”surrogate frames postulate” or when used with reference to standard rods, the ”surrogaterods postulate.”

The first general relativity point above is often a source of confusion, as it is sometimes saidthat the speed of light in general relativity can be different than c. This is true if, for example,one measures the speed of light near a massive star using a clock based on earth. (Time on sucha clock is effectively the coordinate time in a Schwarzchild coordinate system.) As is well known,due to the intense gravitation field, the passage of time close to the star is dilated relative toearth time, and one would indeed calculate a light speed other than c. However, use of standardrods and clocks adjacent the light ray itself would result in a speed of precisely c.

Other confusion exists for scenarios where spacetime itself expands or contracts. For example,just after the big bang, space itself was expanding much like the surface of a balloon being blownup. A photon in space (analogous to an ant on the surface of the balloon) at a different locationthan an observer could then move away from the observer faster than c (analogous to faster thanthe ant can crawl on the surface) because the space (balloon surface) between the photon and theobserver is itself expanding. Yet a photon spatially coincident with an observer could never beseen by that observer to have speed greater than c, and local standard rods and clocks adjacentany photon would find its speed equaling c regardless of the dynamical state of spacetime itself.

1.3 Nomenclature and Definitions

Upper-case letters herein shall refer to inertial systems; lower case to non-inertial systems. Forexample K shall designate the non-rotating (lab) frame; k, the rotating frame.

Flat spacetime will be referred to as ”Minkowski space”; whereas the term ”Minkowskimetric” will be limited to refer only to a Minkowskian set of coordinates (Cartesian plus time)used within that flat space. Hence a Minkowski space need not be represented solely by aMinkowski metric, and we will in fact use cylindrical coordinates, i.e., (cT,R,Φ,Z), for the flatnon-rotating inertial frame.

Though much of the paper may be understood with no working knowledge of differentialgeometry (the mathematics of general relativity), in Sec. 4 it is needed to derive certain results.These derivations are based on a Minkowski metric defined as

ηα,β =

−1 0 0 00 1 0 00 0 1 00 0 0 1

, α, β = 0, 1, 2, 3 (1)

To eliminate confusion for the non-specialist, and to more readily compare results with thoseof Einstein and others, we employ cgs, not geometrized, units where c =2.998 X 1010 cm/sec.

3

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1.4 Rotating Disk Experimental Evidence

1.4.1 The speed of light.

In 1913 Sagnac [16] first demonstrated experimentally that rotating disks exhibit a remarkableproperty, the significance of which the present author believes has been completely overlookedever since. That is, the local speed of a beam of light tangent to the disk circumference is notinvariant, and not isotropic (as seen from the disk).

Fig. 1 depicts the Sagnac experiment schematically. A light beam is emitted radially fromthe center of a rotating disk and is split by a half silvered mirror M at radius r. From there onepart of the beam is reflected by mirrors appropriately placed on the disk such that it travels inone direction around the circumference. The other half of the beam travels the same route overprecisely the same distance, but in the opposite direction. The beams then meet up again andare reflected back to the center where interference of the two beams results in a fringing, i.e., adisplacement of one light wave with respect to the other.

This is exactly the effect Michelson and Morley were first looking for, but could, due to nowwell known relativistic effects, never detect. Fringing results in either experiment would haveimplied different velocities of light in different directions. Hence, while the Michelson-Morleyresult indicated that for translational motion the speed of light is invariant and isotropic, theSagnac experiment indicates that for rotational motion, no such conclusion may be drawn.

The results of Sagnac and others who have repeated his experiment have experimental ac-curacy only to first order in v/c = ωr/c, and indicate that the speed of a light ray tangent tothe circumference measured locally on the disk is equal to [17]

|vlight| ∼= c ± ωr (2)

where the approximately equal sign implies accuracy to first order, and the sign in front of thelast term depends on the relative direction of the rim tangent and light ray velocities.

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These results should, in fact, be expected. An inertial (non-rotating) observer of the twolight rays would see each of them having the speed c, and during the time they are travelingaround the circumference the disk would rotate some amount. Hence one ray would meet backup with the half-silvered mirror M before the other, and an observer fixed on the disk at Mwould conclude that the speeds in each direction were different. Further, the difference can bereadily calculated, to first order, to be that shown in Eq. (2). (Selleri [18] makes the calculationrigorously to all orders.) Still further, the effect is local since angular velocity is constant anddue to symmetry, any segment of a constant radius path (in the ideal experimental design) isequivalent to any other segment. Hence any global (average) speed effect measured by Sagnacover finite times and distances is equal to the local (infinitesimal) speed at any point on thecircumference.

1.4.2 Absolute nature of rotational velocity.

The Michelson-Morley experiment also implied that translational velocities are relative, andthat there is no preferred system of reference (no ”ether”).

Rotational velocities, on the contrary, are absolute. (The term ”absolute” herein impliesaccordance with Mach’s principle, i.e., absoluteness with respect to the distant galaxies, withinEinstein’s relativistic theory of flat spacetime). This is due, at least in part, to the absolutenature of the radially directed accelerations experienced by any rotating object. Hence, forrotational velocities there is a preferred frame, and it is the one in which no radial accelerationsare experienced. Any observer, in any frame, can tell which system is the non-rotating one, i.e.,the ”preferred frame”, and how much each of the other frames is rotating relative to it. Thiscan be done by watching the motion of a Foucault pendulum, by noticing whether or not thereis a Coriolis ”force”, or by a number of other means.

2 DIFFICULTIES WITH THE TRADITIONAL VIEW

2.1 The Postulates

The reader has no doubt noticed that the experimental results of Sec. 1.4 above appear to clashwith the very postulates of Sec. 1.2 upon which the theory of relativity was founded. That is,all of the relativistic behavior with which we have become so familiar in the twentieth century,such as the Lorentz contraction and the lack of agreement on simultaneity, are a direct result ofi) invariance of the speed of light, and ii) ”reference frame democracy” (all frames are equal).Yet, the author contends, these postulates simply do not hold for rotating frames.

This point seems to have been overlooked for two reasons. Firstly, much of the literature [16]covering the Sagnac effect focuses on the fringe effect per se, and its concomitant mathematicaldescription, without noting the significant implications such fringing has for the speed of light.Secondly, of those who were aware that this implied a variable speed of light, most probablyglossed over the fact by assuming that in some manner it was merely another general relativistic(non-inertial systems) manifestation of the ”light speed unequal to c effect”. Yet, unlike theexamples provided in Sec. 1.2, there is no expansion or contraction of spacetime associated withthe rotating disk, and the fringing implies a true difference in the local measurement of lightspeed.

Therefore one should, indeed must, expect rotating frame behavior to differ from that of

5

translational motion. Relativistic effects such as the Lorentz contraction are not given a priori ;they are derived. And they are derived from different empirically based principles than thosegoverning rotational motion. Hence, we should be wary of conclusions drawn by simply applyingderived tenets of relativity theory to rotating disks, as Einstein and others have done.

2.2 Geodesic Deviation

The Riemann curvature tensor (or simply ”Riemann”) R is a measure of the curvature of a givenspace. It is defined by virtue of the geodesic deviation equation of differential geometry [19],

∇u∇un + R(..., u, n, u) = 0 . (3)

The first term above represents the deviation between two geodesics, i.e., the rate of changeof the rate of change in proper distance of the perpendicular from the first geodesic to the secondas one travels along the first. If R is zero, then Eq. (3) dictates that every pair of geodesics whichare initially parallel will stay parallel along their entire length. The proper distance betweenthem will never change, and they will never intersect. Since this is true only of flat spaces, azero value for R means the space is flat. IfR 6= 0, such as on the surface of a globe, two geodesics(e.g., lines of longitude) which start out parallel (at the equator), don’t stay parallel (and crossat the poles). R is characteristic of the space itself, not the coordinate system used within thatspace. If, for example, R = 0 for a 2D flat space with Cartesian coordinate system, when wetransform to a polar coordinate system, we still have R = 0.

It is commonly known [20, 21] that the four-dimensional (4D) spacetime of the rotatingsystem (denoted k) is Riemann flat since R = 0 in the non-rotating frame (denoted K), and therotating system coordinates are obtained by simply transforming the 4D coordinates of K intok. If Riemann is zero in K, it must also be zero in k. However, it does not necessarily followthat the subspace of the disk surface, embedded in the 4D space, is flat. By analogy, the 2Dsubspace surface of a sphere embedded in a flat 3D space is not itself flat.

Particles attached to the rotating disk undergo acceleration and hence do not follow geodesicpaths. The path of a free particle or light ray, however, is a geodesic, and though it is straightas seen from K, it looks curved, even ”corkscrew-like,” as seen from k.

The question of flatness for the subspace of the disk can be addressed by considering twofree particles traveling at the same velocity in K in the plane of the disk surface (i.e., the axialcoordinate Z = constant), and tracing out parallel lines in K. The observer in K sees them asstraight and never intersecting. The rotating observer sees them as corkscrew-like and neverintersecting. The point is that the geodesic equation, from which the Riemann tensor is defined,relates to the ”never intersecting” part, not the ”non-straight” part. In a curved space thereis geodesic deviation. It says two geodesics deviate in their behavior. The two geodesics inquestion do not. Further, the two geodesics travel in the plane of the disk surface. Regardlessof how one wishes to define the disk surface and all of the issues of simultaneity involved (seelater sections), the basic fact remains that for z = constant, the two geodesics do not deviate(i.e., they never cross). For a Riemann curved surface they must deviate. Therefore the disksurface is Riemann flat.

By analogy, two parallel geodesics which appear straight in an inertial system appear curvedto an observer in a rectilinearly accelerating system. But they never appear to cross to theaccelerating observer, and the proper distance between them never changes. As is well known,the space of a rectilinearly accelerating system is flat [22]. This is in full accord with Eq. (3)

6

Spacelike path of AJ integrationcT

X

YA B

JcT2

X2

K2 frame

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since geodesic deviation for such a system is zero, and so is Riemann, even though geodesicsthemselves do not look straight.

Geodesic deviation causes tidal forces, the stretching and compressing of a finite sized objectin free fall (i.e., traveling a geodesic). Gravity tries to make one side of the object accelerate in adifferent direction, or at a different rate, than the other side. But a finite sized object travelingalong a geodesic in the plane of the rotating disk would notexperience any tidal forces, and allobservers, whether on the disk, the lab, or anywhere else, would agree there is no stress or strainwithin the object. Hence Riemann is zero along the path of the object, and the surface of thedisk can not be curved.

2.3 Tangent Frames and the Discontinuity in Time

Applying traditional relativistic concepts directly to the rotating disk leads to another strikingdifficulty. It predicts a discontinuity in time on the surface of the disk, and in addition, thelocation of that discontinuity is arbitrary, being merely a function of the particular predilectionsof the observer. In other words, a continuous standard tape measure extending one circumferencearound the rim would not meet back up with itself at the same point in time. The logic leadingto this conclusion follows.

In order to evaluate disk curvature, prior researchers have invoked the ”surrogate framespostulate” (see general relativity principle 4 of Sec. 1.2) and used a series of inertial referenceframes tangent to the disk rim with velocities equal to that of the rim edge (i.e., with v=ωr). It is argued that since acceleration body forces do not affect standard rod length, rods inthese inertial frames should be affected in precisely the same manner as rods aligned with, andattached to, the edge of the disk rim in k.

The problems with this approach can be illustrated with the aid of Fig. 2. Inertial measuringrods in inertial frames K1 to K8 with speeds ωr can be imagined as shown. For practical reasonswe only show eight finite length rods, and we consider them as a symbolic representation ofan infinite number of rods of infinitesimal length. A and B are events located in space at theendpoints of the K1 rod which are simultaneous as seen from K1; B and C are events located inspace at the endpoints of the K2 rod which are simultaneous in K2; and so on for the other eventsC to J. A,B, ...J can be envisioned as flashes of light emitted by bulbs situated equidistantlyaround the disk rim.

7

p is a spatial (three dimensional) point fixed to the disk at which both A and J occur. q is thespatial point on the disk at which B occurs. In principle, A, B, ... J, as well as p and q arelocated on the disk rim though they may not look so in Fig. 2 since the tangent rods shown arenot infinitesimal in length.

Note that although events A and B are simultaneous as seen from K1, they are not simul-taneous as seen in K (via standard relativity theory for two inertial frames in relative motion).As seen from K, A occurs before B. Similarly, B occurs before C, and so on around the rim. Ifthe events are light flashes, a ground based observer looking down on the disk would see theA flash, then B, then C, etc. Hence we conclude that as seen from K, A occurs before J eventhough A and J are both located at the same 3D point p fixed to the rim. As seen from K,during the time interval between events A and J the disk rotates, and hence the point p moves.(As an aside, Fig. 2 can now be seen to be merely symbolic since events A to J would not inactuality be seen from K to occur at the locations shown in Fig. 2. That is, by the time the Kobserver sees the B flash, the disk has rotated a little. It rotates a little more before he sees theC flash, etc.)

According to the traditional treatment of the rotating disk, one then uses the K i rods andintegrates (adds the rod lengths) along the path AB ...J, moving sequentially from tangentinertial frame to tangent inertial frame. This path is represented by the solid line in Fig. 3, andone can visualize small Minkowski coordinate frames situated at every point along the curve AJ(see K2 in Fig. 3) with integration taking place along a series of spatial axes (such as X2 in Fig.3). By doing this one arrives at a length for AJ, the presumed circumference of a disk of radiusr, of precisely as predicted by Einstein and many others

AJ =2πr

√

1 − ω2r2/c2. (4)

But consider that since point p moves along a timelike path as seen from K (see dotted linein Fig. 3), a time difference between events A and J must therefore exist as measured by aclock attached to point p. As a result, one end of a continuous tape measure riding with therim of the disk would not meet back up with its other end at the same point in time. But anymeaningful measurement of the circumference simply must have the same starting and endingevent, and therefore must be a closed path in spacetime.

We have therefore shown that the tangent frames analysis approach leads to a discontinuityin time, a seemingly impossible physical situation. Even further, the spatial location of thatdiscontinuity is completely arbitrary. It depends on where we choose our initial starting pointp. This is a very serious dilemma for the traditional interpretation.

We conclude that simultaneity can not be defined in a consistent manner using local inertialclocks over any closed path where different parts of the path have different relative velocities.Hence, we are unable to measure the circumference of the rim with local inertial rods where theendpoints of all the rods are simultaneous as measured by local inertial clocks. Therefore, inertialframes tangent to the rim can not be used to measure the disk circumference, and conclusionsmade from so doing will not be valid.

This apparent violation of the heretofore seemingly sacrosanct ”surrogate rods postulate” isaddressed in Sec. 5.2.

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2.4 Related Problems

Ehrenfest [23, 24] saw a paradox in the presumed circumferential Lorentz contraction effectwhich Einstein [25] and Grøn [26] attempted to resolve by claiming that the disk circumferencetries to contract in Lorentz fashion, but can’t, and so undergoes internal tensile stress. Othermechanically induced stresses aside, the disk presumably cannot be spun up to relativistic speedswithout developing such stresses and, at high enough speeds, rupturing.

But one must then also argue that the time discontinuity of Fig. 3 is resisted in some way bya ”tension” in the time component around the circumference. If the rods must be extended inorder to meet up in space, then surely the endpoints of rods must be adjusted in time in orderto meet up as well. Tensile stress may be a well known physical phenomenon in a material bodyin space, but there is certainly no such phenomenon associated with time.

A related problem is pointed out by several authors (see, for example Weber [13]). If light rayssent out around the circumference are used according to the standard Einstein synchronizationprocedure, one finds the clock at p at 360˚ to be out of synchronization with itself at 0˚ . Thisleads to the restriction that one can only consider open paths on the disk surface. But thenone must ask what prevents a physical disk based observer from traveling around one completecircumference? And what prevents her from laying down a continuous tape measure as she doesso? And finally, how good a representation of the physical world is a model in which a clockcan not be synchronized with itself?

3 NEW THEORY OF ROTATING FRAMES

In this section we re-derive key aspects of relativity theory for the rotating reference frame usingtwo new postulates based on the Sagnac and other experiments. We follow logic similar tothat employed by Einstein to derive special relativity for translational motion, but start from adifferent, but equally empirically justifiable, basis.

In Sec. 4 transformation techniques of differential geometry are utilized to rigorously deriveall relevant characteristics of the rotating frame, including the exact form of Eq. (2), our firstpostulate below. The present Sec. 3, on the other hand, provides a physically meaningful, andsimpler, derivation of certain of those characteristics.

3.1 New Postulates

We postulate the following:1. The speed of light is not invariant between the ground and the rotating frame, and in the

rotating frame is found to first order by the velocity addition law of Eq. (2)

|vlight| ∼= c ± ωr

2. Observers can discern which frame is non-rotating (the ”preferred frame”).

3.2 Different Results

3.2.1 Simultaneity.

Fig. 4 depicts a means for defining simultaneity at any radius r on the disk. Light rays can beimagined as emitted simultaneously from the centerpoint of the disk, striking mirrors located at

9

T = 0

T ' > 0

T ' ' > T '

T '' ' > T ' '

AB

marks fromevent B

marks from A

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r, and being reflected back to the centerpoint. They all arrive back at the center at the sameinstant in time as measured by a clock located there, and one concludes that the events occurringwhen the light struck the mirrors are all simultaneous. These events are also simultaneous toobservers in K.

The question then arises as to whether a non-inertial observer riding on the rim itself wouldagree that those same events are simultaneous. Standard relativity theory predicts she wouldnot, since she and the K observer have relative velocity difference.

We answer this question by re-considering Einstein’s famous gedanken experiment of thepassing train shown in Fig. 5. Lightning strikes both ends of the car and leaves marks on bothends plus the ground. These events are A and B. Given the postulate that light has the samevelocity as seen from the train or the ground, and given that both observers can later measurethe distance to the brown marks left by lightning events and determine that each is 1/2 waybetween their respective marks, the train observer concludes that A occurred before B becauseshe saw the A flash of light first. This she concludes because she knows the speed of light fromboth directions is the same for her. The ground observer sees each flash at the same instant andconcludes the two events were simultaneous since the speed of light is also the same for him inboth directions. This is case 1, the standard special relativity result.

For case 2, suppose instead that nature works in Galilean fashion and the light from A travelsfaster than the light from B as seen from the train frame. (|VA| = c + v and |VB| = c - v wherev is the absolute value of the relative velocity between frames.) The train observer still seesthe A flash first, and still later measures the distance to the marks and knows she is 1/2 waybetween them. But now she also knows that the light from A travels faster, so she would expectto see it first. The math is trivial. She concludes that A and B were indeed simultaneous, asdoes the ground observer.

As shown by the Sagnac experiment the speed of light on the circumference of the diskbehaves as in the second case above. (Assume for the present that Eq. (2) is an exact equality.We will resolve the first order approximation issue in Sec. 4.) The observer on the disk knowsshe is rotating, knows she has tangential velocity relative to the inertial frame K, and knowsfrom Eq. (2) the formula for calculating the velocity of light as seen by her (it is direct addition

10

as for the train case 2 above.) She therefore concludes that two spatially proximate events onthe circumference which are simultaneous in the ground frame are also simultaneous to her eventhough she sees one of them occur first.

Hence, whether measured from the center of the disk, or locally at any other point on thedisk, simultaneity in the disk frame k is identical with that of K. (Selleri [18] agrees with thisconclusion, although he takes a different route to get there.) So unlike systems with relativerectilinear velocities where there is no common agreement in simultaneity, systems with relativerotational velocities all do agree on simultaneity.

3.2.2 No Lorentz Contraction.

The Lorentz contraction is a direct result of non-agreement in simultaneity between frames. Ifthere is agreement in simultaneity, there is no Lorentz contraction. To show this we need oneadditional, presumably inviolable, postulate. That is,

3. The proper spacetime length of any path is invariant under any transformation, i.e., it isthe same for all observers.

Hence, for two frames in relative motion (notation should be obvious)

(∆s)2 = − c2(∆t)2 + (∆l)2 = − c2(∆t′)2 + (∆l′)2 (5)

For a rod at rest in the primed frame, an observer in the unprimed frame sees that rod suchthat its endpoints are events which for him occur simultaneously, i.e., ∆t = 0. But in the primedsystem those events are not, according to standard relativity theory, simultaneous and ∆t ′ 6= 0.This means ∆l 6= ∆l′, and results in Lorentz contraction [27].

If, however, the same two events could also appear simultaneous in the primed system, then∆t′ = 0, and ∆lmust equal ∆l′. This is, of course, not possible for two frames in relativetranslational motion, but, as we have shown, it is possible between two frames with differentrotational motion.

Hence, if ∆l′is the length of a (short) standard measuring rod attached to the non-rotatingK frame aligned tangentially to the disk rim, and ∆l is the length of a similar rod attachedtangentially to the rim, then neither rod looks shortened to observers in either frame. There is,therefore, no Lorentz contraction for rotating systems.

The reader should note carefully the distinction here between with the contention of Grøn[10] and others [28] that rods fixed to the disk will not contract since tension in the disk preventsthem from so doing. In contradistinction, we show that there is simply no kinematic imperativefor the rods to try to contract. No tension arises in the disk as it is spun up, and no relativisticallyinduced rupturing occurs.

3.2.3 Time Dilation.

Although frames K and k agree on simultaneity, it can be shown that standard clocks in eachrun at different rates. (Note that two clocks running at different rates can nonetheless bothagree on simultaneity, i.e., that no time elapsed off either one between two events.)

The time dilation effect can be demonstrated with the aid of the spacetime diagram of Fig.6, which shows the helical path of a clock fixed on the disk as seen from frame K and that of

11

X

Y

cT

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a clock fixed in K as seen from K. The moving clock travels the path of 3D point p of Fig. 3extended for one full rotation. The path of that clock is a non-geodesic, while the path of its”twin” fixed in K is a geodesic, a straight line. The proper time passed for each clock is simplyits path length (divided by ic), and this path length can be measured in any frame we choosesince it is frame invariant. We choose frame K since it is the simplest.

In frame K, the proper spatial distance traversed by the disk fixed clock is ∆σ = 2πR =(ω∆T )Rwhere the time interval for one rotation is ∆T.Hence, the proper spacetime path length of themoving clock is

(∆s)2 = − c2(∆τ)2 = −c2(∆T )2 + (∆σ)2 = − c2(∆T )2 + ω2R2(∆T )2 (6)

Hence,

∆τ = ∆t = (1 − r2ω2/c2)1/2∆T = (1 − v2/c2)1/2∆T (7)

and the clock fixed in k on the disk rim runs slower than the K clock on the ground. Also, clocksrun more slowly at greater radii, so it is not possible to synchronize standard clocks at differentradii.

It is noteworthy that similar time dilation effects occur in translationally accelerating sys-tems, yet it is readily shown [22] that such systems are nevertheless Riemann flat. (Accelerationdoes not cause spacetime curvature, gravity does.) Hence, time dilation in and of itself is not asufficient condition for curvature.

Note also that, analogous with the rectilinearly accelerating system, it is not possible tosynchronize standard clocks located at different radii in k since such clocks beat at differentrates.

The analysis of a clock fixed on the disk at a certain radius r is similar to that of the travelingtwin in the classic ”twin paradox”. Both twins live in Minkowski spaces, but the traveling twinfollows a non-geodesic in spacetime (it must decelerate/accelerate to return to earth) and hencehas a shorter elapsed time than the geodesic following sibling. This result, as in the rotating diskcase, is independent of the reference frame of the observer since proper pathlength is invariantunder transformation.

12

4 TRANSFORMATION THEORY

Einstein used his two postulates to derive the Lorentz transformation, from which all relevantrelativistic characteristics may be found. If, conversely, he had known the Lorentz transfor-mation first, he could have then derived his two postulates. In the present sec., we start witha reasonable guess at the correct transformation between rotating frames, analogous to theLorentz transformation between translationally moving frames, and not only derive our originalpostulates, but predict other phenomena as well. As will be shown, these other phenomena areself consistent, do not lead to the difficulties delineated in Sec. 2, and agree with all knownexperiments.

4.1 Rotating Frame Metric and Transformations

Strauss [9], Franklin [29], Trocheries [30], and Takeno [31] have attempted to impose transforma-tions between inertial and rotating frames which make an a priori assumption that the Lorentzcontraction is operative and varies with the radius r of the disk (i.e., varies with the tangentialvelocity in the traditional special relativistic manner). These transformations appear to put thecart before the horse, i.e., they start with the Lorentz contraction built in.

An alternative, and more reasonable transformation (see Eqs. (8a-d) below) found in manysources [10,11,12,13,16,26,[32],[33]] (although with different interpretations and results than thepresent paper) makes no such assumption. It simply makes kinematic connections between thecylindrical rotating and non-rotating coordinate systems which are straightforward and seemmost logical. If the transformation is correct, appropriate effects derivable from it should agreewith experiment, and predicted results should be self consistent.

This coordinate transformation, where upper case coordinates represent the inertial frameK, lower case denote the rotating frame k, and the axis of rotation is coincident with both theZand z axes, is

cT = ct (8a)R = r (8b)Φ = φ + ωt (8c)Z = z (8d)

(8)

ω is the angular velocity of the disk, and t, the coordinate time for the rotating system, isthe proper time of a standard clock located at the origin of the rotating coordinate frame, i.e.,it is equivalent to any standard clock at rest in K. Note that t is only a coordinate. It is merelya label and cannot be expected to equal proper time at any given point on the disk (except, ofcourse, at r= 0).

Assumptions upon which transformation (8) is based are: (i) The radial distance r measuredin k can not be contracted as seen from K since velocity is always perpendicular to R, hence R= r. (ii) Radii in k (i.e., lines of constant φ and constant z) each are straight lines as seen fromeither k or K, move with rotational velocity ω, and are independent of r. (iii) The rotation hasno effect on measurements in the direction of the axis of rotation, i.e., the Z direction, since, likethe radial distance, Z = z is perpendicular to velocity. Assumptions (i) and (iii) are apparentlyuniversally accepted by others. Assumption (ii) leads to Eq. (8c) and, as mentioned, has beenconsidered by others.

13

The transformation (8) seems Galilean in nature, rather than relativistic, and if it is valid(as most researchers today feel that it is), we should not be surprised to find the disk exhibitingat least some Galilean characteristics.

To deduce the metric for the rotating system we begin with the line element for the standardcylindrical coordinate system of the Minkowski space K

ds2 = − c2dT 2 + dR2 + R2dΦ2 + dZ2. (9)

Finding dT, dR, dΦ, and dZ from Eqs. (8), and inserting into Eq. (9), one obtains themetric of the coordinate grid in k. (Note this step incorporates postulate 3 of Sec. 3.2.2, i.e., dsis invariant.)

ds2 = −c2(1 − r2ω2

c2)dt2 + dr2 + r2dφ2 + 2r2ωdφdt + dz2

= gαβdxαdxβ ,

(10)

where the covariant form of the metric gαβ and its inverse, the contravariant matrix gαβ , readilyfound via the standard cofactor method, are

gαβ =

−(1 − r2ω2

c2 ) 0 r2ωc 0

0 1 0 0r2ωc 0 r2 0

0 0 0 1

gαβ =

−1 0 ωc 0

0 1 0 0ωc 0 (1 − r2ω2

c2 )/r2 00 0 0 1

. (11)

For future reference, the comparable matrices in K are

GAB =

−1 0 0 00 1 0 00 0 R2 00 0 0 1

GAB =

−1 0 0 00 1 0 00 0 1

R2 00 0 0 1

(12)

where sub and superscripts A and B as used here are upper case Greek letters for alpha andbeta.

Note from Eqs. (10) and Eqs. ((11) that the rotating disk system is not orthogonal (themetric is not diagonal).

Taking the differentials in Eqs. (8), one can readily derive the matrix ΛαB which transforms

contravariant components of vectors and tensors from K to k, and its inverse ΛAβ which transforms

contravariant vectors and tensors from k to K. These transformations between the two cylindricalcoordinate systems are:

ΛαB =

1 0 0 00 1 0 0−ω

c 0 1 00 0 0 1

, ΛAβ =

1 0 0 00 1 0 0ωc 0 1 00 0 0 1

. (13)

With the above metrics and transformations forming the basis of the new theory, we canproceed to derive the effects we would expect to see in the physical world.

14

4.2 Galilean Characteristics

4.2.1 Invariance of simultaneity.

From Eq. (8a) [or equivalently by comparing Eq. (9) and Eq. (10)], if ∆T =0 in K for twoevents, then ∆t = 0 in k for the same two events. In other words the two events are simultaneousas seen from either system, in agreement with our earlier thought experiment based on physicalreasoning. Note that the transformation between time coordinates of the present theory is muchdifferent than that of the Lorentz transformation. In the latter the coordinate time difference isdependent upon the locations of the two events; in the former, it is not.

4.2.2 No Lorentz contraction.

Note since Eq. (9) equals Eq. (10), the circumference of the disk for any radius r = R, atfixed time tand constant z(i.e., dt = dT = dr = dR = dz = dZ = 0), is 2πr(∆Φ = ∆φ = 2π),implying that the rotating disk is indeed a flat space, and corroborating the physical reasoningof Sec. 3.2.2. (Some authors, most notably Grøn [10,26], contend that the non-time orthogonalnature of the rotating coordinate system negate this conclusion. We resolve this matter in Sec.5 below and the Appendix.)

Note further that Lorentz contraction arises directly from the Lorentz transformation, yetEqs. (8), the transformation now accepted as correct by virtually everyone in the field, is notthe Lorentz transformation. There is, therefore, absolutely no reason (other than tradition) totacitly assume that it must somehow give rise to Lorentz contraction.

4.2.3 Angular velocity addition.

Consider three co-axial reference frames, one of which is not rotating and designated by K, thesecond of which has rotational velocity ω2 and designated by k2, and the third of which hasvelocity ω3 = 2ω2 and is designated by k3. ω2 and ω3 are measured relative to K. Note thatan observer in k2 sees the k3 system rotate once relative to him, for each time interval that herotates once relative to K. Hence ω3/2, the angular velocity of k3 relative to k2, has the samemagnitude as ω2, and therefore

ω3 = ω2 + ω3/2 (14)

Relationship Eq. (14) obviously holds in general, and demonstrates that rotational velocitiesfor co-axial systems add directly, in Galilean fashion, frame to frame and not relativistically as dotranslational velocities. This not only lends further credence to the Galilean type transformation(8) employed herein, but also implies that there is no upper limit on angular velocity comparableto the luminal limitation on rectilinear velocities [34].

4.2.4 Translational velocity addition.

Assume V I are the components of the three velocity of an object as seen in the K cylindricalcoordinate system, and UA are the components of the four-velocity. For the same object, anobserver in k measures vi as components of the three velocity and uα for the four-velocity. Thatis,

15

V I =dXI

dT, UA =

dXA

dτ= 1√

1−v2/c2

cdRdT = V R

dΦdT = V Φ

dZdT = V Z

. (15)

To find the three velocity addition law, we use the same procedure employed in specialrelativity to derive the relativistic velocity addition law. We begin by first transforming thefour-vector dX I to its counterpart in k, dxα, using the appropriate transformation matrix fromEqs. (13).

cdtdrdφdz

= dxα = ΛαBdXB =

1 0 0 00 1 0 0−ω

c 0 1 00 0 0 1

cdTdRdΦdZ

=

cdTdR−ωdT + dΦdZ

(16)

Three velocities in k are then found simply by dividing the spatial components of Eq. (16)by dt, and noting that dt = dT, i.e.,

vi =dxi

dt=

dRdt

−ω dTdt + dΦ

dtdZdt

=

V R

−ω + V Φ

V Z

(17)

For an object with purely tangential velocity of magnitude V Tang equal to RV Φ one findsfrom Eq. (17) that

vtang = rvφ = − ωR + RV Φ=−ωR + V tang (18)

a very Galilean-looking transformation.It must be noted once again, however, that time derivatives above are with respect to coor-

dinate time t,and for a disk fixed observer at any location other than r = 0, time dilation effectsmust be taken into account to reflect the actual velocities such an observer would measure withphysical instruments. In practice this would mean dividing Eq. (18) by the factor

√

1 − ω2r2/c2,i.e., by the factor local time differs from the coordinate time used in Eq. (17). Note that thisdoes not change the directly additive quality of Eq. (18).

4.2.5 Lack of invariance of the speed of light.

Consider Eq. (18) where V Tang represents the speed c of a light ray which could be propagatingin the positive or negative Φ direction. Then

vlight,tang = −ωR ± c (19)

This result is in remarkable agreement with the Sagnac experiment, and provides strongsupport for the validity of transformation (8).

Since velocities in Eq. (19) are coordinate velocities, we must divide both sides of theequation by the time dilation factor

√

1 − ω2r2/c2 to represent the physical velocities a diskobserver would actually measure using local standard clocks. By doing this we obtain the exactrelationship for which the Sagnac result Eq. (2) was only a first order approximation. Hence, as

16

we assumed in Sec. 3.2.1, the exactly equal sign in Eq. (2) is correct if the velocities are takenas those which would actually be measured by an observer fixed to the disk; see Sec. 5.1 andEq. (33).

By utilizing these velocities, rather than c, for light rays employed to synchronize clocks ata given radius, one then finds a clock at 360o is synchronized with itself at 0o. More generally,closed path integrations are fully allowable, and thereby consistent with what one would expectphysically.

Note that we have derived our rotating frame postulates from transformation (8). Eq. (19)is the first postulate. By taking ω = 0 in Eq. (19), we get our second. That is, the preferredframe is the one with isotropic light speed c, i.e., it is the inertial one.

In Sec. 5.1 we re-derive these results even more rigorously, and reconcile them with generalrelativity principle 1 in Sec. 1.2.

4.3 Lorentzian Characteristics

A plethora of cyclotron experiments demonstrates that rotating systems do indeed possess cer-tain relativistic characteristics, such as time dilation (longer decay times) and mass-energy in-crease with speed. If transformation (8) is the correct one, then these effects must be predictedby it. The ensuing derivations do indeed confirm that transformation (8) is consistent with theseexperiments.

4.3.1 Time dilation.

From the metric of Eq. (10) with r, φ, and z constant and ds 2 = −c2dτ2, the proper time atany radius r is

dτ =√

1 − r2ω2

c2 dt =√

1 − v2

c2 dT (20)

which corroborates the result Eq. (7) of Sec. 3.2.3. (Since speed is constant, finite ”∆”differences in Eq. (7) can be taken over to differentials ”d”.)

Note that the time dilation effect arises naturally from the simple and readily justifiablecoordinate transformation (8), and was not ”built in” from the start by assuming that it holds apriori. Further, time dilation does occur in the rotating frame in accordance with the standardrelation of special relativity. However, unlike special relativity this effect is not symmetricbetween frames k and K. Observers in both systems agree that the rotating disk clocks runslower.

4.3.2 Path lengths of light and particles.

The pathlength of any object traveling in spacetime is invariant between frames in accordancewith our ”new” postulate 3, which is, of course, not really new but a fundamental principle ofdifferential geometry. The path length of light, in particular, remains null as viewed from therotating frame since ds = 0 in K, and hence ds must also = 0 in k.

4.3.3 Four-vectors.

The four-velocity and the four-momentum transform readily between the rotating and non-rotating systems also in accordance with basic principles of differential geometry/general rela-

17

tivity.However, when making general transformations of four-vectors, one should keep two things

in mind which are usually irrelevant for Minkowski metrics in Minkowski space, but are quiterelevant for other metrics such as that of the rotating frame. Both of these relate to physicalinterpretation of the components of four-vectors (i.e., the quantities one would actually measurewith instruments.)

The first of these concerns lies with the covariant or contravariant nature of the components.Since coordinate differences (e.g., dx α) are expressed as contravariant quantities, and since four-velocity is simply the derivative of these coordinate differences with respect to the invariantscalar quantity τ (proper time), four-velocities only represent (proper) time derivatives of thecoordinate values if they are expressed in contravariant form. In general, lowering the index ofuα via the metric gαβ gives components uα which are not the time derivatives of the coordinatevalues. This is true because gαβ is not the identity matrix. Note that in inertial frames gαβ

= ηαβ (see Eq. (1)) which is, apart from the sign of the g00 component, an identity matrix.In a coordinate frame with such a Minkowski metric the covariant form of the four-velocityis identical to the contravariant form except for the sign of the timelike component. In othercoordinate frames, however, the difference is much more significant, and care must be taken towork with the contravariant form of the four-velocity.

Four-momentum, on the other hand, must be treated in terms of its covariant components.This is because said four-momentum is the canonical conjugate of the four-velocity. In brief, ifthe Lagrangian of a given system is

L = L(xα, xα, τ) (21)

where dots over quantities represent derivatives with respect to τ , then the conjugate momentumis

pα =∂L

∂xα(22)

Hence it is imperative that one use the covariant components of the four-momentum. Con-travariant components, for all but a Minkowski metric, will not represent physical quantitiessuch as energy, three momentum, etc.

Getting the correct contravariant or covariant components is not quite enough, however, inorder to compare theoretical results with measured quantities. If a given basis vector does nothave unit length, the magnitude of the corresponding component will not equal the physicalquantity measured. For example, a vector with a single non-zero component value of 1 in acoordinate system where the corresponding basis vector for that component has length 3 doesnot have an absolute (physical) length equal to 1, but to three.

In general, therefore, (see Malvern [35], for example, for further explication), physical com-ponents are found from vector components via the relations

vα = vα√gαα vα = vα√

gαα (23)

where carets over indices designate physical quantities, and underlining implies no summation.Hence in order to compare theoretical component values with experiment, it is necessary

to use contravariant components for coordinate differences and four-velocities, covariant compo-nents for four-momenta, and physical components of all component quantities whether covariantor contravariant.

18

4.3.4 Mass-energy of a particle fixed on disk.

Consider a particle of mass m fixed on the disk at constant φ, r, and z. Since dφ = dr = dz =0, the four-momentum of the particle in k coordinate contravariant components (using metricEq. (10)) is

pβ = muβ = mdxβ

dτ= m

c dtdτ

000

=m

√

1 − ω2r2

c2

c000

(24)

where dt/dτ is found from Eq. (20).The mass-energy (non-physical), except for a factor −c, and three momenta (non-physical)

are the four-dimensional conjugate momenta of the dxβ and are the components of the covariantfour-momentum vector

pα = gαβpβ =

−(1 − r2ω2

c2 ) 0 r2ωc 0

0 1 0 0r2ωc 0 r2 0

0 0 0 1

mc√

1−ω2r2

c2

1000

= mc√

1−ω2r2

c2

−(1 − r2ω2

c2 )0r2ωc

0

(25)

The physical energy eof the particle as measured in k is therefore

e = −p0c = −p0

√

−g00c = −p0c = mc2√

1 − ω2r2

c2

= mc2 − 12mω2r2 − (higher order terms)

= mc2 + Vclass + ...... .

(26)

where Vclass is the classical potential for the particle as seen from the rotating frame, and Vclass

plus the higher-order terms is the relativistic potential energy.The energy of the particle as seen from K can be found by using the second of Eqs. (13) to

transform pβ of Eq. (24) into the four-momentum of the inertial frame P B. By then using GAB,the metric of K, PA is found to be

PA = GABPB =

−1 0 0 00 1 0 00 0 R2 00 0 0 1

mc√

1 − ω2R2

c2

10ωc0

(27)

The mass-energy E of the particle as measured from K is therefore

19

E = −P0 c = −P0

√G00c = −P0c = mc2

√

1−ω2R2

c2

= mc2√

1−v2

c2

= mc2 + 12mω2R2 + ....... .

(28)

in full accord with the relativistic mass-energy effect. Note that the total energy in both k andK becomes imaginary at r = c/ω where the tangential disk speed reaches that of light. Thatthese results were obtained from the transformations (8) (i.e., Eqs. (13) derived without ad hocLorentz factors thrown in, supports the contention that those transformations are indeed thecorrect ones.

Note also that PΦ turns out to be the relativistic angular momentum, the conjugate momen-tum of Φ, as it must be if the transformation employed is correct.

PΦ =mR2ω

√

1 − ω2R2

c2

=mvR

√

1 − v2

c2

(29)

Further, pφ, the relativistic angular momentum as seen from k (see Eq. (25)), has the samevalue as PΦ, and is non-zero even though the four-velocity component uφ in k is zero.

5 RAMIFICATIONS OF NON-TIME-ORTHOGONALITY

5.1 The Speed of Light

Consider the line element of Eq. (10) for a ray of light directed tangentially at radius r (withvelocity c in K). dz = dr = 0, and

ds2 = 0 = − c2(1 − ω2r2

c2 )dt2 + r2dφ2 + 2r2ωdφdt . (30)

Solving Eq. (30) for dφ via the standard quadratic equation formula and dividing the resultby dt, one obtains

dφ

dt= − ω ± c

r, vtang =

rdφ

dt= − rω + c . (31)

The same result as Eq. (19).For velocities in terms of local times on the rim, substitute

dt =dtl

√

1 − ω2r2

c2

, (32)

where dt l is time as measured by local standard clocks at r, and hence

vtang,phys =− rω ± c√

1 − ω2r2

c2

(33)

is the exact expression for the first-order approximation Eq. (2) of the Sagnac experiment.Note that without the off diagonal (non-orthogonal) terms of the metric in Eq. (30) the

physical velocity above measured by local standard clocks would be c. (Delete the last term inEq. (30) and substitute the relation Eq. (32) for dt in terms of dt l.)

20

N

M

cT

ct (= cT)

cTK1

XK1

L

∆T = ∆t

∆T = ∆t23

23

∆T = ∆t13

13

c∆Τ= c∆t

v

c

lR∆Φ = r∆φ

^Figure 7. Speed of Light in Three Reference Frames at r = R

Fig. 7 helps to explain this effect of non-orthogonality graphically. The coordinate axesshown as perpendiculars represent K; the slanted coordinate lines represent the local inertialframe at r, K1; the line MN represents the null path of a light ray; and the bold lines representthe non-orthogonal coordinate axes for k at the location r. Note that the coordinate time axesof k and K1 are coincident, as are the spatial axes of k and K. Coordinate times T (in K) and t(in k) are equal (see dashed horizontal lines representing different values of cT = ct). Physicalspatial distance in K (= R∆Φ) and in k (=r∆φ) are equal, and we consider ∆ values as small.Coordinate systems for K and K1 are orthogonal in Minkowski space, the coordinate system fork at r is not. In all three systems the path length of MN is zero, since pathlength is invariant.

Observe that for a given amount of coordinate time (which is the same in both k and K, i.e.,c∆T = c∆t), the light ray travels a certain spatial distance lin k, but a greater spatial distanceL in K. Hence the speed of light measured in k is less than that in K, and this corresponds withthe plus sign before the c in Eq. (31). For a light ray in the opposite direction (minus sign inEq. (31)) one can show graphically (with a light ray MN/ in the second quadrant of Fig. 7 atright angles to MN) that the corresponding l distance is greater than L and hence the velocityfor that ray would be greater in k than in K.

Given that the slope of MN is unity, L = c∆T . Dividing this by ∆T , one gets the speed oflight in K as c. The k time axis has slope c/v = c/ωr, so l = L - c∆T (ωr/c). Dividing this by∆T , one arrives at Eq. (31) for the coordinate speed of light in k (with the plus sign for c sincelight ray MN is traveling in the direction of disk rotation).

Note that in both inertial frames K and K1, the speed of light ray MN equals c. We cantherefore conclude that general relativity principle 1 remains valid, provided we constrain it torefer to time orthogonal frames (such as the Minkowski, Schwarzchild, and Friedman geometries).It does not hold for non-time-orthogonal frames.

21

A Bka

KV= 0

Va= 0

a

Fig. 8. Equivalent Inertial and Non-Inertial Rods

5.2 The ”Surrogate Rods Postulate”

Non-time-orthogonality also reconciles the results of Sec. 2.3 (tangent frames can not be used tomeasure the circumference) with the heretofore seemingly universal applicability of the surrogaterods postulate of Sec. 1.2. We note first, however, that the tangent frames do not, strictlyspeaking, have the same velocity as the disk rim. The rim segment, in addition to its linearvelocity component v = ωr, has an angular velocity ω which the tangent frame does not. Hence,unlike other successful applications of the surrogate rods postulate, the tangent frames here donot mimic the rim frame velocity in all regards. Therefore, they can not, in the truest sensebe considered ”co-moving” as many prior researchers have assumed. To see the effect of thisin terms of non-time-orthogonality, we first consider the underlying principles on which thesurrogate rods postulate is based.

Fig. 8 shows two spatially coincident standard rods with zero relative velocity, the first ofwhich is fixed in a rectilinearly accelerating frame ka, and the second of which is fixed in aninertial frame K. Consider two light flash events A and B located at the endpoints of both rods.An observer in K at the centerpoint of the inertial rod sees both flashes at the same time andconcludes they were simultaneous as seen from the K frame. Likewise, an observer on the ka rodhalfway between A and B would see them at the same time as well and know that the eventswere also simultaneous as seen from the ka frame. That is, ∆ta = ∆T = 0 between A and B.Since the proper spacetime length between the two events is the same as seen from both frames(i.e., ∆sa = ∆S), then the spatial length between them must also be equal, and the lengthmeasured by rods in ka and K between A and B are equal. Similar arguments can be made foracceleration in the rod lengthwise direction, as well as for gravitational body forces induced bya massive body.

Hence the surrogate rods postulate is merely a restatement of the proper spacetime pathlength invariance postulate for the special case where ∆(time) = 0 for both observers (which itis for zero relative velocity and time orthogonal frames). Figs. 9 and 10 reveal what this meansin the context of non-time-orthogonal reference frames.

Fig. 9 depicts two inertial reference frames, K and K1, in relative motion, and serves as areview of the cause of the Lorentz contraction effect. A rod fixed in K1 has lengthL1 =L asseen from K1. The endpoints (and all points between) ofL1 move with velocity vrelative to Kalong world lines parallel to the K1 time axis. Lorentz contraction arises because the observerin K sees rod endpoint events as A and C, whereas the observer in K1 sees them as A and B.L1’, the distance between A and C is less thanL, the distance between A and B, by the Lorentz

22

B

A

K1 spatial axis

L (= L1)L1' K spatial axisC

L1

cT cTK1

L1 = L > L1'

Fig. 9. Standard Rods inTwo Inertial Frames

DL in K K spatial axis

E k spatial axis

cT ct (= cT)

lk in k

L = lk

Fig. 10. Standard Rod in Non-time-orthogonal Frame

contraction factor. Though it is beside the point we are in the process of making, the Lorentzcontraction effect is thus seen to be little more than an optical illusion fostered on us by lack ofagreement in simultaneity. No Lorentz contracted object ever feels compressed.

In contrast with Fig. 9, Fig. 10 shows the rotating coordinate frame k at radius r superim-posed with the non-rotating inertial frame K. In Fig. 10 we show two circumferentially alignedstandard rods, one designated by lk which rides with the disk and the other, L, fixed in K. Whenboth rods are at rest in the same inertial system, they have equal length, i.e., lk = L. Whenthe disk is spinning they also have equal length, as seen by both k and K observers, since, asdiscussed earlier, the same endpoint events (D and E) of both rods are seen as simultaneousin both frames. Yet, due to the special nature of the non-time-orthogonal frame k at r, the lkrod has a non-zero velocity relative to the L rod. Every point on the lk rod in Fig. 10 moveswith the same velocity as every point on the L1 rod in Fig. 9. Yet L1 looks contracted from K,whereas lk does not, i.e.,

L 1 = L = lk > L′1 (34)

That is, even though L1 and lk have the same velocity as seen from K, they do not have thesame spatial length as seen from K. Prior researchers have almost universally assumed they do.

That L1 looks contracted as seen from k can be corroborated by superimposing the k frameof Fig. 10 with the K1 frame of Fig. 9. Hence, two rods with the same velocity, one in atime orthogonal frame and one in a non-time-orthogonal frame do not have equal lengths. Weconclude that the surrogate rods postulate is only valid for time orthogonal frames.

5.3 The Large Radius, Small ω Limit

Several researchers [13,18] have considered the limiting case of very large radius, small angularvelocity, with large circumferential velocity v = ωr. In this case acceleration v2/r approacheszero, and it is argued there is no way to discern between such a frame and an inertial frame.Hence, a circumferential segment of the rotating frame in such case must approximate an inertial(Lorentzian) frame.

The answer to this conundrum lies in non-time-orthogonality. From Fig. 7 it can be seenthat the slope of the time axis in the k frame is c/v, and as we have shown, the Sagnac effect, the

23

lack of Lorentz contraction, and all other peculiarities of the rotating frame are derivable fromnon-time-orthogonality, i.e., the slope of that axis. But in taking the limit described, v remainsconstant, and hence so does the slope of the time axis. Therefore, all of the non-Lorentzianphenomena heretofore described for rotating frames are unmitigated in passing to the limit.(See also Sec. 6.)

Contrary to what many claim, an observer on this limiting case frame can determine she isrotating. In fact, three experiments can reveal this. The first is the Sagnac experiment. Thesecond is described in Section 6.2 below. The third involves measuring the mass of a knownentity such as an electron which varies relativistically with the potential energy, i.e., as a functionof v2 alone as in Eq. (26), and hence one can readily determine v.

6 THE NEW THEORY AND EXPERIMENT

6.1 Michelson-Morley Revisited

Given the speed of our planet around its sun, and the speed of our solar system around itsgalactic center, one might ask why measurements on our planet (which could be consideredas part of a frame rotating about the center of each of these systems) do not seem to exhibitthe aforementioned non-Lorentzian properties. In particular, why did Michelson and Morleynot find the speed of light in the direction of galactic rotation different from that in otherdirections? Note that given the Sagnac results, this question has an empirical imperative whichis independent of any theory, i.e., any particular rotating frame analysis.

The answer, the author submits, is that bodies in gravitational orbits follow geodesics, i.e.,they are in ”free fall”. That is, they are in locally inertial frames and therefore obey Lorentzianmechanics. Objects fixed in ”true” rotational frames, on the other hand, are held in place bynon-gravitational forces, do not travel geodesic paths, and exhibit Sagnac type characteristics.Hence, the only effective rotational velocity for the earth is the earth surface velocity about itsown (inertial) axis. Michelson and Gale [36] did in fact measure the Sagnac effect for the earth’ssurface velocity in the 1920’s. And in order to be maintain accuracy, the Global PositioningSystem must apply a Sagnac velocity correction to its electromagnetic signals [37].

6.2 Modern Michelson-Morley Experiments

The most significant experiment, however, and the most accurate Michelson-Morley type testto date is that of Brillet and Hall [38]. They found a ”null” effect at the ∆t/t= 3X10−15 level,ostensibly verifying standard relativity theory to high order. However, to obtain this resultthey subtracted out a persistent “spurious” signal of amplitude 2X10−13 at twice the apparatusrotation frequency.

Compare this anomalous signal to that predicted by the presently proposed theory. Thevelocity of the earth surface at 40o latitude, where the Brillet and Hall experiment was performed,is .355 km/sec. If the speed of light is truly increased or decreased by this amount in the directionof rotation, then a Michelson-Morley experiment (see Fig. 11) with one leg in the direction ofthe velocity would yield [39]

∆t

t∼= 1

2

v2

c2(35)

24

Figure 11. Michelson-Morley Experiment

velocity

10 20 30 40 50 60 70 80 90100110120130140150170180190200mm 876543210i nches

l

l

Figure 12. Modified Michelson-Morley Experiment with Decreased Sensitivity

velocity

10 20 30 40 50 60 70 80 90100110120130140150170180190200mm 876543210i nches

l

l

d

d

where tis the round-trip time for one leg and ∆t is the difference in time taken between the legaligned with the velocity vector and the leg perpendicular to that vector.

Brillet and Hall rotated their equipment about a vertical axis (always perpendicular to theearth surface velocity). In such a case Eq. (35) would yield the peak-to-peak amplitude of thesignal. To compare with Brillet and Hall’s reported single peak amplitude signal, one must thendivide Eq. (35) by two. Using .355 km/sec in Eq. (35) and dividing by two, one gets ∆t/t=3.5X10−13. Fig. 12 can help to explain the reason for the discrepancy in this number and thereported value.

In Fig. 12 the two light rays no longer travel solely on two perpendicular paths. In theBrillet and Hall experimental apparatus a similar configuration to that of Fig. 12 was used,apparently to accommodate the laser equipment which provided such extraordinary accuracy.(See Fig. 1 in Brillet and Hall.) Note that if the two shorter legs dwere each 25% the length ofthe primary legs l, then the time t in Eq. (35) would be 1.25 times that of Fig. 11. Note alsothat the lower of the d legs is aligned perpendicularly to the velocity, whereas the lower path isintended to monitor light speed in the direction of the velocity vector. If both it and the upperd leg were 25% of the primary legs in length, one could therefore expect a 25% reduction in ∆tas well. Hence the total signal ∆t/twould be a reduced by a factor of .75/1.25 = 60% .

Although Brillet and Hall did not provide the pertinent dimensions in their paper, fromthe sketch of their equipment one can conclude that the 25% figure used above may be fairlyaccurate. To account for this approximation we can assume a signal strength modification factorof 50% to 70% . For these percentages, the expected signal range is 1.7-2.5X10−13, in remarkableagreement with the measured value of 2X10−13.

Note also that in the Brillet and Hall experiment signals were exchanged electronically be-tween the two legs, and that actual fringing at a single location was not measured. Hence, wemay expect the variation in light speed to affect these transmission signals as well, introducingadditional error.

The author also investigated possible mitigating effects from ”frame dragging” (a tφ offdiagonal term appearing in the metric due to the earth’s angular momentum), and the chordpath effect (the light ray parallel to the velocity actually travels a chord of the arc length, notthe arc length itself). These were found to be negligible to orders of magnitude well beyond10−13.

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For completeness, we also mention the results of Hils and Hall [40] which found no variationsto an order of 2X10−13 (only coincidentally the same number as above.) However, the Hils andHall apparatus was fixed relative to the earth’s surface and hence was immune to variationsof the type predicted by the present theory (which would manifest only if the apparatus wererotated relative to the earth surface.)

6.3 A Proposed Experiment

We propose another experiment using a combined Sagnac and Michelson-Morley type apparatusin order to further test the present theory. For this experiment light is emitted from a rotatingdisk center in the manner of the Sagnac experiment, but a Michelson-Morley type apparatus ismounted on the disk rim. When the light from the center reaches the rim it is split into twocomponents. One of these travels along the circumference a short distance and then is reflecteddirectly backwards (rather than further out around the rim.) The other component of the lightis reflected in the z direction (perpendicular to the disk) an equal distance and reflected directlybackwards as well. The two returning components are then deflected toward the disk centerwhere fringe effects are measured. Accelerating the disk, and accounting for elastic deformation,one should find the degree of fringing varies. The standard theory predicts no such variation.

7 ELECTRODYNAMICS, MECHANICS, AND SPACETIME

With regard to electrodynamics, Ridgely [33] has recently used covariant constitutive equationsin an elegant analysis to answer a troubling question cogently posed by Pellegrini and Swift [32].He uses transformation (8) to derive electrodynamic results for the rotating frame k, not thetangent frames Ki, and finds that those results match what one would find by simply applyingMaxwell’s equations and traditional special relativity to the tangent frames.

We can conclude the following. Only with the theoretical approach shown herein can oneobtain internally consistent results which agree with all experiments. However, for the purposesof mass-energy, momentum, and time dilation calculations (shown herein) and Maxwell’s equa-tions (shown by Ridgely), one can get away with assuming that the tangent frames representthe rotating frame and using traditional special relativity. That is, in these cases the laws ofnature conspire to make both the present and the traditional analysis produce the same resultfor observers in K (i.e., mass-energy dependence on ωr, electric polarization, etc.). When itcomes to matters of time (synchronization, simultaneity), space (curvature), and

Michelson-Morley/Sagnac type experiments, however, then analysis must be confined to therotating frame itself, else the inconsistencies of Sec. 2 and inexplicable “spurious” experimentalsignals inevitably arise.

It therefore appears that the rotating disk problem may have, at long last, been completelysolved. According to Ridgely’s results and the theory proposed herein, no paradoxes remain andall theory matches up with the physical world as we know it.

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8 SUMMARY AND CONCLUSIONS

8.1 New Theory Predictions

The lack of Lorentz contraction, agreement in simultaneity, flatness of the disk surface, non-invariant/non-isotropic speed of light, and time dilation can all be derived in two different ways:(i) directly from transformation (8); and (ii) from the Sagnac experiment. All but the asteriskedphenomena summarized below can be determined in at least two ways (i.e., transformation (8),experiment, or thought experiment based on Sagnac). An asterisk (*) indicates the conclusiondepends only on the validity of transformation (8).

1. The subspace surface of a rotating disk is flat.2. No circumferential Lorentz contraction exists. Further, no relativistically induced tensile

stress develops as there is no kinematic imperative for the disk circumference to try to Lorentzcontract.

3. Observers anywhere in the rotating frame and observers in the non-rotating frame allagree on simultaneity.

4. Velocities (angular and translational*) add directly frame-to-frame and not relativistically.5. Angular velocities are absolute and have no upper speed limitation.6. Rods in inertial frames with velocities equal to the tangent velocities at a given disk radius

can not be used to measure the circumference, sincea. the ”surrogate frames postulate” for equivalence of inertial and non-inertial standard rods isnot valid for the rotating frame, and is generally invalid for any non-time-orthogonal frame, andb. doing so leads to a discontinuity in time.

7. Light has a null path length, yet the local speed of light in the rotating frame (and allnon-time-orthogonal frames) is not isotropic and generally not equal to c.

8. Time dilation does occur, but it is not symmetric, i.e., rotating and non-rotating observersagree that time dilation occurs on the disk relative to the stationary frame.

9. A particle fixed on the disk exhibits relativistic mass-energy dependence on tangentialvelocity. (No asterisk since cyclotron experiments validate this effect.)

10. Only the theory proposed herein yields self consistent results which completely conformwith physical reality. However, use of tangent frames and traditional special relativity producethe same results for a certain subset of phenomena.

8.2 Comparison of Various Perspectives

The proposed theory resolves all difficulties with the traditional disk analysis delineated in Sec.2.

Both the proposed theory and the traditional analysis agree with cyclotron experiments,i.e., they both predict time dilation (longer particle decay times), as well as relativistic mass-energy dependence on speed. Both theories are also consonant with the Phipps [41] experimentwhich has been used to discount certain other prior approaches to the problem not based ontransformation (8). Importantly, however, the new theory predicts the results of the Brillet andHall [38] experiment which the standard theory does not.

The new theory also agrees with part of Grøn’s first work [42] on the standard approach,where he uses transformation (8) and concludes from it that simultaneity on the disk and in thelab are the same. However his latter paper [10] employs tangent frames analysis and recounts

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purported difficulties in accelerating the disk which are the direct result of disagreement insimultaneity. In the presently proposed theory no such kinematic restriction on disk accelerationexists, and there is no mechanism by which any such disk would rupture from relativisticallyinduced tensile stress, as has been contended by Einstein, Grøn, and others.

Table I summarizes the similarities and differences between the approaches of Einstein, Grøn,and the proposed theory.

TABLE I. COMPARISON OF VARIOUS DISK ANALYSES

Einstein Grøn ThisPaper

Postulates agree with experiment? No No Yes

Discontinuity in time? Yes Yes & No No

Clock synchronized with itself? No No Yes

Closed paths allowable? No No Yes

”Tension” in time required? Yes Yes No

Predicts Brillet & Hall anomaly? No No Yes

Agrees with cyclotron experiments? Yes Yes Yes

Relativistic mass-energy? Yes Yes Yes

Time dilation on disk? Yes Yes Yes

Lorentz contraction effect? Yes Yes No

Disk surface is curved? Yes Yes No

Relativistically induced disk stress? Yes Yes No

Same simultaneity: disk and lab? No Yes & No Yes

Time as defined is observable?* No No Yes

Transformation is Galilean type? No Yes Yes

Restricts surrogate rods principle? No No Yes

Speed of light = c on disk? Yes No† No

Agrees with Phipps experiment? Not treated Yes Yes

*See Appendix† Result derived from transformation (8), but effect on relativity postulates not considered.

ACKNOWLEDGMENT. The author would like to express his sincerest gratitude toRobin Ticciati and Arthur Swift for reading early versions of the manuscript, for offering valu-able suggestions, and above all, for their encouragement and support for this work, despite itsparadigm-challenging nature.

1 APPENDIX: THE METRIC OF PRIOR TREATMENTS

Landau and Lifshitz [12], Møller [11], Strauss [9], and Grøn [10] have all discussed a threedimensional submetric of the four-dimensional rotating frame metric defined by

28

γij = gij − g0ig0j

g00. (A1)

So defined, γij represents the spatial metric which is locally orthogonal to the local propertime axis. Some of these authors have then used this metric to determine whether the space ofthe rotating disk is flat or not, and have concluded that it is curved. In fact, using the metricof Eq. (11) for gαβ in Eq. (A1) one finds

γij =

1 0 0

0 r2

1−r2ω2

c2

0

0 0 1

(A2)

A line element around the circumference then becomes

ds = r√

1−r2ω2

c2

dφ = r√

1−v2

c2

dφ (A3)

From Eq. (A3) it is obvious that the circumference Cis not equal to 2πr,and in fact is equalto what Einstein and some of the above authors have claimed.

Further, Riemann for the metric of Eq. (A2) is non-zero.However, the metric Eq. (A2) is derived for a differential line element having simultaneous

starting and ending points as measured by local inertial clocks. In other words it assumesthat a local inertial frame is aligned with ds and that measurement is carried out such that theendpoints of ds are simultaneous in that inertial frame. But this is nothing other than the type ofintegration path we investigated herein with the aid of Fig. 3 (solid line). As demonstrated, suchan integration cannot be carried out around a closed path on the surface of the disk wherein thestarting and ending points are simultaneous, and hence, it is meaningless as a physical measureof the disk circumference.

Adler, Bazin, and Schiffer [43] use the time transformation

dt∗ = dt − ωr2

c2−ω2r2 dφ (A4)

in Eq. (10) and obtain the same metric Eq. (A2), where the new coordinate time is then t*.However, the transformation Eq. (A4) is like any transformation in that it effectively shifts

one to a different reference frame. Hence the time t* as defined no longer represents time onthe rotating frame itself, but some other time. This other time definition, we contend, has nomeaning in the sense of being actually observable in the physical world by any possible observer.In essence, it represents an observer who miraculously skips from tangent inertial frame totangent inertial frame without the concomitant acceleration and rotation associated with the

29

disk itself. Not only is this not possible, but such a definition of time leads to a temporaldiscontinuity, as we have shown.

REFERENCES AND NOTES

References

[1] Stachel, J., ”Einstein and the Rigidly Rotating Disk”, Chapter 1 in Held, General Relativityand Gravitation (Plenum Press, New York, 1980), pp. 1-15.

[2] Einstein, A., Relativity, The Special and General Theory: A Popular Exposition (Methuenand Co., Ltd., London, 1920).

[3] Einstein, A., The Meaning of Relativity (Princeton University Press, Princeton, N.J., 1921).

[4] Einstein, A., and Infeld, L., The Evolution of Physics (Simon and Schuster, 1938), pp.226-234.

[5] Einstein, A., ”Die grundlage der allgemeinen relativitatstheorie,” Ann. Phys. (Leipzig), 49,769-822 (1916).

[6] Ref. –1, pp. 7.

[7] Ref. –1, pp. 8.

[8] Levy, H., Modern Science, a Study of Physical Science in the World Today, (A. Knopf, NewYork, 1939), pp. 595.

[9] Strauss, M., ”Rotating Frames in Special Relativity”, Int’l Journal of Theoretical Physics,11, 107-123 (1974).

[10] Grøn, Ø., ”Rotating Frames in Special Relativity Analyzed in Light of a Recent Article byM. Strauss”, Int’l Journal of Theoretical Physics 16(8), 603-614 (1977).

[11] Møller, C., The Theory of Relativity, (Oxford at the Clarendon Press, 1969), pp. 223.

[12] Landau, L.D., and Lifshitz, E.M., The Classical Theory of Fields, (Addison-Wesley, Read-ing, MA 1962), pp. 271-298.

[13] T.A. Weber, “Measurements on a rotating frame in relativity, and the Wilson and Wilsonexperiment”, Am. J. Phys. 65 (10), 946-953 October 1997.

[14] This is strictly true only, as we shall see, for time orthogonal frames. For such frames thisis equivalent to saying light always travels null paths.

[15] Ref. –1, pp. 9.

[16] Post, E.J., ”Sagnac Effect”, Mod. Phys. 39, 475-493 (1967).

[17] Post (Ref. –16) presents the Sagnac results in terms of the fringe shift (his equation (1)).Using (2) of the present paper, one can derive (1) in Post (use A = π r 2), and vice versa.

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[18] Selleri, F., “Noninvariant One Way Speed of Light and Locally Equivalent ReferenceFrames”, Found.Phys. Lett., 10, 73-83 (1997)

[19] Misner, C.W., Thorne, K.S., and Wheeler, J.A., Gravitation, (W.H. Freeman and Co., NewYork 1973), pp. 275.

[20] Ref. –1, pp. 8.

[21] Ref. –9, pp. 107.

[22] Ref. –19, pp. 173. Using the metric for rectilinearly accelerating systems one can imme-diately see that time is dilated along the direction of the acceleration. Calculation of theRiemann tensor from the metric results in zero value for Riemann for the 4D space of theaccelerating system. Since the metric is diagonal, any subspace of two or more coordinatesis also Riemann flat.

[23] Ehrenfest, P., ”Gleichfomrige Rotation starrer Korper und Relativitatheorie”, PhysikalischeZeitschrift 10, 918-918 (1909).

[24] Sama, N., ”On the Ehrenfest Paradox”, Am. J. Phys. 40, 415-418 (1972).

[25] Ref. –1, pp. 9.

[26] Grøn, Ø., ”Relativistic Description of a Rotating Disk”, Am. J. Phys 43(10), 869-876(1975).

[27] There is a little subtlety here. Calculation of the Lorentz contraction actually involves rodendpoints which appear to be different events to different observers. The example assumesthe same two events are measured by both observers. However, the conclusion remainsvalid. If the endpoint events look simultaneous to two different observers, then the rodlength measured must be the same for each.

[28] Arzelies, H., Relativistic Kinematics, (Pergamon Press, New York 1966), Chapter IX.Arzelies was not convinced that tension arises in the disk, but references others who were.

[29] Franklin, P., ”The Meaning of Rotation in the Special Theory of Relativity”, Proceedingsof the National Academy of Sciences of the USA 8(9), 265-268 (1922).

[30] Trocheries, M.G., ”Electrodynamics in a Rotating Frame of Reference”, Philosophical Mag-azine 40(310), 1143-1154 (1949).

[31] Takeno, H., ”On Relativistic Theory of Rotating Disk”, Progress in Theoretical Physics7(4), 367-376 (1952).

[32] Pellegrini, G.N, and Swift, A.R., ”Maxwell’s Equations in a Rotating Medium: Is There aProblem?, Am. J. Phys. 63(8), 694-705 Aug 1995.

[33] Ridgely, C.T., “Applying relativistic electrodynamics to a rotating material medium”, Am.J. Phys. 66 (2) 114-121 February 1998.

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[34] As will be seen, tangential velocity still can not exceed c, and angular velocity of a disk ofany given radius will thereby be limited to a maximum value. However, a disk of arbitrarilysmaller radius could spin at arbitrarily greater angular velocity.

[35] Malvern, L.E., Introduction to the Mechanics of a Continuous Medium, (Prentice-Hall,Englewood Cliffs, New Jersey, 1969), Appendix I, Sec. 5, pp. 606-613.

[36] Michelson, A.A., and Gale, H.G., “The Effect of the Earth’s Rotation on the Velocity ofLight, Part II”, The Astrophysical Journal 61, 140-145 (1925). See also Michelson, A.A.,”The Effect of the Earth’s Rotation on the Velocity of Light, Part I”, The AstrophysicalJournal 61, 137-139 (1925).

[37] Allan, D. W. and Weiss, M. A., “Around-the-World Relativistic Sagnac Experiment,” Sci-ence, 228, 69-70 (1985.)

[38] Brillet, A. and Hall, J.L., “Improved Laser Test of the Isotropy of Space”, Phys. Rev. Lett.,42, 9, 549-552 (February 26, 1979.)

[39] Born, M., Einstein’s Theory of Relativity, revised edition (Dover Publications, Inc., NewYork 1965), pp. 216.

[40] Hils, D., and Hall, J.L., ”Improved Kennedy-Thorndike Experiment to Test Special Rela-tivity”, Phys. Rev. Lett., 64, 15, 1697-1700 (April 9, 1990.)

[41] Phipps, T., ”Kinematics of a Rigid Rotor”, Nuovo Cimento Letters 9, 467-470 (1974)

[42] Ref. –26, pp. 873

[43] Adler, R., Bazin, M., and Schiffer, M., Introduction to General Relativity, (McGraw-Hill,New York, 1975), pp. 124.

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